Question

1. solve the following by factoring. be sure to state the zeros/x - intercepts of the relation.

a) (9n^{2}-25) b) (4x^{2}+20x=-25)

1. use the discriminant to determine the number of real solutions for each quadratic relation.

a) (5x^{2}-4x + 11=0) b) (x^{2}=8x - 16)

1. in business, a company breaks even when its total costs and total revenues are equal. this means the business has not lost any money, nor has it made any profit.

a camping supplies company models the profit of its newest backpack according to the function (p=-3x^{2}+36x - 81), where (x) is the number of backpacks sold, in hundreds, and (p) is the profit, in thousands of dollars. how many backpacks must the company sell to break - even? do not use quadratic formula.

Explanation:

Step1: Solve 9n² - 25 by factoring

Use difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\). Here \(a = 3n\) and \(b = 5\), so \(9n^{2}-25=(3n + 5)(3n - 5)\). Set each factor equal to zero: \(3n+5 = 0\) gives \(n=-\frac{5}{3}\), and \(3n - 5=0\) gives \(n=\frac{5}{3}\). The zero/x - intercepts are \(n =-\frac{5}{3}\) and \(n=\frac{5}{3}\).

Step2: Solve 4x² + 20x=-25

Rewrite it in standard form \(4x^{2}+20x + 25 = 0\). Notice it's a perfect - square trinomial \((2x + 5)^{2}=0\). Set \(2x+5 = 0\), then \(x=-\frac{5}{2}\). The zero/x - intercept is \(x =-\frac{5}{2}\).

Step3: Find discriminant of 5x² - 4x + 11 = 0

The discriminant formula is \(\Delta=b^{2}-4ac\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here \(a = 5\), \(b=-4\), \(c = 11\). So \(\Delta=(-4)^{2}-4\times5\times11=16-220=-204\lt0\), so there are no real solutions.

Step4: Find discriminant of x²=8x - 16

Rewrite in standard form \(x^{2}-8x + 16 = 0\). Here \(a = 1\), \(b=-8\), \(c = 16\). Then \(\Delta=(-8)^{2}-4\times1\times16=64 - 64=0\), so there is 1 real solution.

Step5: Solve for break - even in business problem

The company breaks even when \(P = 0\), so \(-3x^{2}+36x - 81 = 0\). Divide through by \(-3\) to get \(x^{2}-12x + 27 = 0\). Factor it as \((x - 3)(x - 9)=0\). Set each factor equal to zero: \(x-3 = 0\) gives \(x = 3\), and \(x - 9=0\) gives \(x = 9\). Since \(x\) is in hundreds, the company must sell 300 or 900 backpacks to break - even.

Answer:

a) For \(9n^{2}-25\), zero/x - intercepts are \(n =-\frac{5}{3}\) and \(n=\frac{5}{3}\); for \(4x^{2}+20x=-25\), zero/x - intercept is \(x =-\frac{5}{2}\)
b) For \(5x^{2}-4x + 11 = 0\), no real solutions; for \(x^{2}=8x - 16\), 1 real solution
c) The company must sell 300 or 900 backpacks to break - even.